Mon., Dec. 6:
# In fact, 2s_1/2 and 2p_1/2 are not exactly degenerate. Various
effects contribute to the
breaking of this degeneracy, e.g. nuclear finite size effects, nuclear
magnetism, relativistic corrections
to the nuclear motion, and retardation, but the largest by far is the
interaction with the dyanmical
fluctuations of the quantum electromagnetic field, i.e. the QED
effects. The splitting of this
level was suspected by some from ambiguous spectroscopic evidence,
clouded by poor resolution
due to finite temperature Doppler broadening, until in 1947 Lamb and
Retherford established
with certainty that there is an upward shift of the 2s_1/2 state by
about 1000 MHz, about 1/10
of the (2s_1/2, 2p_1/2) --- 2p_3/2 fine structure splitting. (Later
refinements showed it to be
close to 1058 MHz). It is now called the Lamb shift. The method
used by Lamb and collaborators
involved molecular beams and RF technology which was an outgrowth of
Rabi's magnetic
resonance method, which decreased the effects of Doppler broadening.
# Lamb reported his results at the famous Shelter Island Conference
on the foundations of
quantum mechanics, which was attended by all the physics luminaries
of the time, including
Bethe, Schwinger, and Feynman. The Lamb shift shocked people into trying
harder to get
a finite result from QED, which on the face of things predicts
an infinite shift. The general
attitude at the time had been that whatever was infinite must be zero,
but now it was evidently
not zero. On the train on the way back from the meeting, Bethe managed
to get a finite result,
and one that agreed roughly with the measurements, by a crude nonrelativistic
approximation.
He noted that the shift of the energy of a FREE electron would also
be infinite, so he subtracted
out this part, reasoning that the observed electron mass must be a
sum of the "bare mass"
and this infinite correction. This still left an infinite result, but
one that is only logarithmically
divergent in the momentum cutoff, and therefore not very sensitive
to the precise value
of the cutoff. Bethe chose for the cutoff the value mc, reasoning that
the relativisitc theory
will give something like this as an effective cutoff. And it does.
# The QED origin of the Lamb shift can be thought of as arising from
an extra jiggling of the
electron due to its interaction with the vacuum fluctuations of the
electromagnetic field,
with and average squared amplitude of (2alpha/pi) ln(1/alpha
)(hbar/mc)^2. As in the explanation
of the Darwin term, if we average the Coulomb potential over a region
of this size we find to
lowest order no change except at the origin where the Laplacian of
1/r is nonzero. Thus s-waves
feel this effect most strongly, and the effect is to raise the s-levels,
since the potential is effectively
not as deep as it would otherwise have been. In fact there is also
a small effect for nonzero
angular momentum orbitals.
# Zeeman effect: Splitting of levels in a uniform external
magnetic field.
H_Z = -mu.B = (mu_B B/hbar)(L_z + 2S_z) = (mu_B B/hbar)(J_z + S_z).
Recall mu_B = 5.8 x 10^-5 eV/T.
For s-states this gives an energy shift 2 mu_B B m_s, m_s = +/-
1/2.
For nonzero L, we must either use perturbation theory, with either
H_Z or H_s.o. the peturbation,
depending on whether mu_B B is small or large compared with the spin-orbit
splitting,
or diagonalize the spin-orbit interaction together with the Zeeman
term if neither effect is
dominiant. At about one Tesla the two effects are of the same order.
# For example consider the weak field Zeeman shifting of the
2p_3/2 levels. H_Z is diagonal
in the degenerate subspace in the basis |3/2,m_ j>. This is obvious
for the J_z term. The S_z
term doesn't touch the spatial wf, and for different m_j of the spatial
wf has the same m_l then the
m_s must be different, so S_z is also diagonal in this basis of the
degenerate states. Thus the
contribution of the S_z term to the first order shifts is just obtained
by taking the expectation value
of S_z in the unperturbed states. This we did with the help of the
table of Clebsch-Gordon
coefficients derived last week.
# To use perturbation theory for the strong field Zeeman effect, we
use the basis states
|lm_lm_s>. The Zeeman term is diagonal in this basis, so the shift
is just (mu_B B/hbar)(m_l+ 2m_s).
The remaining degeneracy is in states with the same value of m_l +
2m_s.
To evaluate the spin-orbit perturbation to this write S.L =
S_z L_z + (S_+ L_- + S_- L_+)/2. The
first term is diagonal in the unperturbed basis, and the term in parentheses
preserves m_l + m_s,
but changes m_s, hence it has no matrix elements at all between states
of the same m_l + 2m_s.
The spin-orbit shift is therefore proportional to m_s m_l.
# To treat the intermediate field effect, note that both spin-orbit
and Zeeman terms commute
with J_z, so they do not mix states of different m_j. The states |j=3/2,
m_ j = +/- 3/2>
are eigenstates of both terms, so the shift is trivial to evaluate.
The states |j=3/2, m_ j =1/2>
and |j=1/2, m_ j=1/2> do mix under the Zeeman term, so we must diagonalize
a 2 x 2 matrix here
(or at least find the eigenvalues, and similarly for the states |j=3/2,
m_ j =-1/2> and |j=1/2, m_ j=-1/2>.
# In hw#12.8 you evaluate the weak, strong, and intermediate Zeeman
shift for a p-orbital.
Wed. Dec. 8:
Lamb shift experiment
Hyperfine hamiltonian
Fri. Dec. 10:
Hyperfine splitting
Zeeman of hyperfine
review